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2025
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2024
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2023
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2022
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2021
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2020
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2019
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2017
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2015
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sample_questions 7
2019 national-I-science_gkztc

14 maths questions

Q3 Probability Definitions Set Operations View
3. After the examination ends, submit both this test paper and the answer sheet. Section I: Multiple Choice Questions: This section has 12 questions, each worth 5 points, for a total of 60 points. For each question, only one of the four options is correct.
1. Given sets $M = \{ x \mid - 4 < x < 2 \} , N = \left\{ x \mid x ^ { 2 } - x - 6 < 0 \right\}$ , then $M \cap N =$
A. $\{ x \mid - 4 < x < 3 \}$
B. $\{ x \mid - 4 < x < - 2 \}$
C. $\{ x \mid - 2 < x < 2 \}$
D. $\{ x \mid 2 < x < 3 \}$
2. Let complex number $z$ satisfy $| z - \mathrm { i } | = 1$ , and the point corresponding to $z$ in the complex plane is $( x , y )$ , then
A. $( x + 1 ) ^ { 2 } + y ^ { 2 } = 1$
B. $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$
C. $x ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
D. $x ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
3. Given $a = \log _ { 2 } 0.2 , b = 2 ^ { 0.2 } , c = 0.2 ^ { 0.3 }$ , then
A. $a < b < c$
B. $a < c < b$
C. $c < a < b$
D. $b < c < a$
Q5 Curve Sketching Identifying the Correct Graph of a Function View
5. The graph of function $f ( x ) = \frac { \sin x + x } { \cos x + x ^ { 2 } }$ on $[ - \pi , \pi ]$ is approximately
A. [Figure]
B. [Figure]
C. [Figure]
D. [Figure]
Q6 Combinations & Selection Combinatorial Probability View
6. In ancient Chinese classics, the ``Book of Changes'' uses ``hexagrams'' to describe the changes of all things. Each ``hexagram'' consists of 6 lines arranged from bottom to top, with lines divided into yang lines ``——'' and yin lines ``——'', as shown in the figure. If a hexagram is randomly selected from all hexagrams, the probability that it has exactly 3 yang lines is
A. $\frac { 5 } { 16 }$
B. $\frac { 11 } { 32 }$
C. $\frac { 21 } { 32 }$
D. $\frac { 11 } { 16 }$
Q7 Vectors Introduction & 2D Angle or Cosine Between Vectors View
7. Given non-zero vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 2 | \boldsymbol { b } |$ and $( \boldsymbol { a } - \boldsymbol { b } ) \perp \boldsymbol { b }$ , the angle between $\boldsymbol { a }$ and $\boldsymbol { b }$ is
A. $\frac { \pi } { 6 }$
B. $\frac { \pi } { 3 }$
C. $\frac { 2 \pi } { 3 }$
D. $\frac { 5 \pi } { 6 }$
Q9 Arithmetic Sequences and Series Find General Term Formula View
9. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ . Given $S _ { 4 } = 0 , a _ { 5 } = 5$ , then
A. $a _ { n } = 2 n - 5$
B. $a _ { n } = 3 n - 10$
C. $S _ { n } = 2 n ^ { 2 } - 8 n$
D. $S _ { n } = \frac { 1 } { 2 } n ^ { 2 } - 2 n$
Q10 Circles Circle Equation Derivation View
10. Given that the foci of ellipse $C$ are $F _ { 1 } ( - 1,0 ) , F _ { 2 } ( 1,0 )$ , and a line through $F _ { 2 }$ intersects $C$ at points $A , B$ . If $\left| A F _ { 2 } \right| = 2 \left| F _ { 2 } B \right|$ and $| A B | = \left| B F _ { 1 } \right|$ , then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$
B. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
C. $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$
D. $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$
Q11 Trig Graphs & Exact Values View
11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$ , there are four conclusions:
(1) $f ( x )$ is an even function
(2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$
(3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$
(4) The maximum value of $f ( x )$ is 2
The numbers of all correct conclusions are
A. (1)(2)(4)
B. (2)(4)
C. (1)(4)
D. (1)(3)
Q13 Chain Rule Chain Rule with Composition of Explicit Functions View
13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$ .
Q14 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
14. Let $S _ { n }$ denote the sum of the first $n$ terms of a geometric sequence $\left\{ a _ { n } \right\}$ . If $a _ { 1 } = \frac { 1 } { 3 } , a _ { 4 } ^ { 2 } = a _ { 6 }$ , then $S _ { 5 } =$ $\_\_\_\_$ .
Q15 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
15. Teams A and B are in a basketball championship series using a best-of-seven format (the series ends when one team wins four games). Based on previous results, Team A's home/away schedule is ``home, home, away, away, home, away, home'' in order. Team A's probability of winning at home is 0.6, and away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$ .
Q16 Polar coordinates View
16. Given hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 1 }$ intersects the two asymptotes of $C$ at points $A , B$ respectively. If $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B }$ and $\overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$ , then the eccentricity of $C$ is $\_\_\_\_$ .
Section III: Solution Questions: Total 70 points. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22 and 23 are optional; choose one to answer.
(I) Required Questions: Total 60 points
Q17 12 marks Trig Proofs Triangle Trigonometric Relation View
17. (12 points) In $\triangle A B C$ , let the sides opposite to angles $A , B , C$ be $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$ .
(1) Find $A$ ;
(2) If $\sqrt { 2 } a + b = 2 c$ , find $\sin C$ .
Q22 Polar coordinates View
22. Solution: (1) Since $-1 < \frac{1-t^2}{1+t^2} \leq 1$ and $x^2 + \left(\frac{y}{2}\right)^2 = \left(\frac{1-t^2}{1+t^2}\right)^2 + \frac{4t^2}{(1+t^2)^2} = 1$, the rectangular coordinate equation of $C$ is $x^2 + \frac{y^2}{4} = 1$ $(x \neq -1)$. The rectangular coordinate equation of $l$ is $2x + \sqrt{3}y + 11 = 0$.
(2) From (1) we can set the parametric equation of $C$ as $\left\{\begin{array}{l} x = \cos\alpha, \\ y = 2\sin\alpha \end{array}\right.$ ($\alpha$ is the parameter, $-\pi < \alpha < \pi$). The distance from a point on $C$ to $l$ is $\frac{|2\cos\alpha + 2\sqrt{3}\sin\alpha + 11|}{\sqrt{7}} = \frac{4\cos\left(\alpha - \frac{\pi}{3}\right) + 11}{\sqrt{7}}$.
When $\alpha = -\frac{2\pi}{3}$, $4\cos\left(\alpha - \frac{\pi}{3}\right) + 11$ attains its minimum value of 7, therefore the minimum distance from a point on $C$ to $l$ is $\sqrt{7}$.
Q23 3 marks Proof Direct Proof of an Inequality View
23. Solution: (1) Since $a^2 + b^2 \geq 2ab$, $b^2 + c^2 \geq 2bc$, $c^2 + a^2 \geq 2ac$, and $abc = 1$, we have $a^2 + b^2 + c^2 \geq ab + bc + ca = \frac{ab + bc + ca}{abc} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. Therefore $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq a^2 + b^2 + c^2$.
(2) Since $a, b, c$ are positive numbers and $abc = 1$, we have $(a+b)^3 + (b+c)^3 + (c+a)^3 \geq 3\sqrt[3]{(a+b)^3(b+c)^3(a+c)^3}$ $= 3(a+b)(b+c)(a+c)$ $\geq 3 \times (2\sqrt{ab}) \times (2\sqrt{bc}) \times (2\sqrt{ac})$ $= 24$. Therefore $(a+b)^3 + (b+c)^3 + (c+a)^3 \geq 24$.